Associative Lie algebra without Jacobi identity

Question

1) Is there a name for associative Lie algebra that does not require Jacobi identity to hold?

2) Can such algebra exist, and if it does exist, can this algebra contain infinitely many elements?

3) Regardless of whether number of elements is finite or not, can anyone present an example of how ordinary integers or vectors can be used to define such algebra?

Answer 1

This is simply called an anticommutative associative algebra. Any module can be given this structure by giving it the zero product: $[x,y] = 0$ for all $x,y$. Other examples include exterior algebras $\Lambda(x_1, x_2, \dots)$. Such an algebra can contain infinitely many elements (give $\mathbb{Z}$ the zero product).

Answer 2

1.) The name is skew-symmetric associative algebra, or anti-commutative associative algebra.

2.) The class of associative Lie algebras consists exactly of nilpotent Lie algebras of class $c\le 2$. Indeed, a Lie algebra is associative if and only if $[x,[y,z]]=[[x,y],z]]$ for all $x,y,z$. In other words, we have $[x,[y,z]]+[z,[x,y]]=0$. By the Jacobi identity this means $[y,[z,x]]=0$ for all $x,y,z$, i.e. the Lie algebra is abelian or $2$-step nilpotent.

We can define classes of algebras by requiring identities, like commutativity, associativity, anticommutativity, Jacobi identity, and so on. There are all combinations possible. We may consider anticommutative associative algebras, or commutative associative algebras, Leibniz algebras (just Jacobi identity but no anticommutativity). By the way, commutative algebras satisfying the Jacobi identity are called Jacobi-Jordan algebras. They are quite different from Lie algebras in general.